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The critical temperature for the BCS equation at weak coupling. (English) Zbl 1137.82025

In this paper it is studied the Bardeen-Cooper-Schrieffer (BCS) equation for a Fermi gas at real chemical potential \(\mu\) and positive temperature \(T\), with local two-body interaction \(\lambda V(x)\), where \(\lambda\) denotes a coupling constant and \(v\in L^1(\mathbb R^3)\cap L^{3/2}(\mathbb R^3)\). The main result of the present paper establishes the asymptotic behaviour of the critical temperature as \(\lambda\) approaches 0. There are also obtained necessary and sufficient conditions on \(V(x)\) for the existence of a nontrivial solution for all values of \(\lambda >0\). The approach developed in this paper is not restricted to the kinetic energy \(K_{T,\mu}\) appearing in the BCS model, but it can be adopted to any symbol vanishing on a manifold of codimension at least one.

MSC:

82D55 Statistical mechanics of superconductors
46N50 Applications of functional analysis in quantum physics
82D50 Statistical mechanics of superfluids
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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